A commutative Bezout \(PM^{\ast}\) domain is an elementary divisor ring

B. V. Zabavsky, A. Gatalevych

Abstract


We prove that any commutative Bezout \(PM^{\ast}\) domain is an elementary divisor ring.

Keywords


Bezout domain, PM-ring, clean element, neat element, elementary divisor ring, stable range 1, neat range 1

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References


J. W. Brewer, P. F. Conrad, P. R. Montgomery. Lattice-ordered groups and a conjecture for adequate domains, Proc. Amer. Math. Soc, 43(1) (1974), pp.31–34. pp.93–108.

M. Contessa, On pm-rings, Comm. Algebra, 10(1) (1982), pp.93–108.

G. De Marco, A. Orsatti. Commutative rings in which every prime ideal is contained in a unigue maximal ideal, Proc. Amer. Math. Soc, 30(3) (1971), pp.459–466.

M. Henriksen, Some remarks about elementary divisor rings, Michigan Math. J., 3(1955/56) pp.159-163.

O. Helmer, The elementary divisor for certain rings without chain conditions, Bull. Amer. Math. Soc., 49(2)(1943), pp.225–236.

I. Kaplansky. Elementary divisors and modules, Trans. Amer. Math. Soc., 66(1949), pp. 464–491.

M. Larsen, W. Lewis, T. Shores, Elementary divisor rings and finitely presented modules, Trans. Amer. Mat. Soc. 187(1974) pp.231–248.

W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229(1977) pp. 269–278.

W. McGovern, Neat rings, J. of Pure and Appl. Algebra, 205(2)(2006) pp. 243–266.

B.V. Zabavsky, Diagonal reduction of matrices over rings, Mathematical Studies, Monograph Series, v. XVI, Lviv (2012), 251p.

B.V. Zabavsky, Diagonal reduction of matrices over finite stable range, Mat. Stud., 41(1)(2014) pp.101–108.

B.V. Zabavsky, Questions related to the K-theoretical aspect of Bezout rings with various stable range conditions, Mat.Stud. 42(1)(2014), pp. 89–109.


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